SB ESS 755
A SERIES OF PROBLEMS,
FAMILIARIZE THE PUPIL WITH GEOMETRICAL CONCEPTIONS^
AND TO EXERCISE HIS INVENTIVE FACULTY.
WILLIAM GEORGE SPENCER,
WITH A PREFATORY NOTE
BY HERBERT SPENCER.
NEW YOFK . : - CINCINNATI - : CHICAGO :
AMERICAN BOOK COMPANY
HUXLEY'S INTRODUCTORY VOLUME.
FOSTER AND TRACY'S PHYSIOLOGY AND
GEIKIE'S PHYSICAL GEOGRAPHY
HUNTER'S HISTORY OF PHILOSOPHY.
LUPTON'S SCIENTIFIC AGRICULTURE.
SPENCER'S INVENTIONAL GEOMETRY.
JEVONS'S POLITICAL ECONOMY.
TAYLOR'S PIANOFORTE PLAYING.
PATTON'S NATURAL RESOURCES OF THE
WENDEL'S HISTORY OF EGYPT.
FREEMAN'S HISTORY OF EUROPE.
FYFFE'S HISTORY OF GREECE.
CREIGHTON'S HISTORY OF ROME.
MAHAFFY'S OLD GREEK LIFE.
WILKINS'S ROMAN ANTIQUITIES.
TIGHE'S ROMAN CONSTITUTION.
ADAMS'S MEDIAEVAL CIVILIZATION.
YONGE'S HISTORY OF FRANCE.
BROOKE'S ENGLISH LITERATURE.
WATKINS'S AMERICAN LITERATURE.
ALDEN'S STUDIES IN BRYANT.
MORRIS'S ENGLISH GRAMMAR.
MORRIS AND BOWEN'S ENGLISH GRAM-
NfCHQL'S ENGXL^H JCGty POSITION.
JEBB'S GREEK LITERATURE.
GLADSTONE'S HOMER. ,
$ fJLAS'SIC AL GEOGRAPHY.
SPENCER INV. GEOM.
COPYRIGHT, 1876, BY D. ArPLETON & CO.
w. P. 13
WHEN it is considered that by geometry the
architect constructs our buildings, the civil en-
gineer our railways ; that by a higher kind of
geometry, the surveyor makes a map of a
county or of a kingdom ; that a geometry still
higher is the foundation of the noble science of
the astronomer, who by it not only determines
the diameter of the globe he lives upon, but as
well the sizes of the sun, moon, and planets,
and their distances from us and from each
other ; when it is considered, also, that by this
higher kind of geometry, with the assistance of
a chart and a mariner's compass, the sailor navi-
gates the ocean with success, and thus brings all
nations into amicable intercourse it will surely
be allowed that its elements should be as acces-
sible as possible.
Geometry may be divided into two parts-
practical and theoretical : the practical bearing
a similar relation to the theoretical that arith-
metic does to algebra. And just as arithmetic
is made to precede algebra, should practical ge-
ometry be made to precede theoretical geome-
Arithmetic is not undervalued because it is
inferior to algebra, nor ought practical geome-
try to be despised because theoretical geometry
is the nobler of the two.
However excellent arithmetic may be as an
instrument for strengthening the intellectual
powers, geometry is far more so ; for as it is
easier to see the relation of surface to surface
and of line to line, than of one number to an-
other, so it is easier to induce a habit of reason-
ing by means of geometry than it is by means
of arithmetic. If taught judiciously, the collat-
eral advantages of practical geometry are not
inconsiderable. Besides introducing to our no-
tice, in their proper order, many of the terms
of the physical sciences, it offers the most favor-
able means of comprehending those terms, and
impressing them upon the memory. It educates
the hand to dexterity and neatness, the eye to
accuracy of perception, and the judgment to
the appreciation of beautiful forms. These ad-
vantages alone claim for it a place in the educa-
tion of all, not excepting that of women. Had
practical geometry been taught as arithmetic is
taught, its value would scarcely have required
insisting on. But the didactic method hitherto
used in teaching it does not exhibit its powers
Any true geometrician who wjll teach practi-
cal geometry by definitions and questions there-
on, will find that he can thus create a far great-
er interest in the science than he can by the
usual course ; and, on adhering to the plan, he
will perceive that it brings into earlier activity
that highly-valuable but much-neglected power,
the power to invent. It is this fact that has in-
duced the author to choose as a suitable name
for it, the inventional method of teaching prac
He has diligently watched its effects on both
*exes, and his experience enables him to say
that its tendency is to lead the pupil to rely on
his own resources, to systematize his discoveries
in order that he may use them, and to gradually
induce such a degree of self-reliance as enables
him to prosecute his subsequent studies with sat-
isfaction: especially if they st'onld happen to
be such studies as Euclid's "Elen-ents," the nse
of the globes, or perspective.
A word or two as to using the definitions
and questions. Whether they relate to the
mensuration of solids, or surfaces, or of lines ;
\vhether they Belong to common square meas-
ure, or to duodecimals ; or whether they apper-
tain to the canon of trigonometry ; it is not the
author's intention that the definitions should be
learned by rote ; but he recommends that the
pupil should give an appropriate illustration oi
each as a proof that he understands it.
Again, instead of dictating to the pupil how
to construct a geometrical figure say a square
and letting him rest satisfied with being able
to construct one from that dictation, the author
has so organized these questions that by doing
justice to each in its turn, the pupil finds that,
when he comes to it, he can construct a square
The greater part of the questions accompany-
ing the definitions require for their answers ge-
ometrical figures and diagrams, accurately con-
structed by means of a pair of compasses, a scale
of equal parts, and a protractor, while others
require a verbal answer merely. In order to
place the pupil as much as possible in the state
in which Nature places him, some questions
have been asked that involve an impossibility.
Whenever a departure from the scientific
order of the questions occurs, such departure
has been preferred for the sake of allowing
time for the pupil to solve some difficult prob-
lem ; inasmuch as it tends far more to the for-
mation of a self-reliant character, that the pupil
should be allowed time to solve such difficult
problem, than that he should be either hurried
The inventive power grows best in the sun
shine of encouragement. Its first shoots are
tender. Upbraiding a pupil with his want of
skill, acts like a frost upon them, and materially
checks their growth. It is partly on account of
the dormant state in which the inventive power
is found in most persons, and partly that very
young beginners may not feel intimidated, that
the introductory questions have teen made so
TO THE PUPIL
WHEN it is found desirable to save time, omit
copying the definitions ; but when time can be
spared, copy them into the trial-book, to im-
press the terms on the memory.
In constructing a figure that you know, use
arcs if you prefer them ; but, in all your at-
tempts to solve a problem, prefer whole circles
to arcs. Circles are suggestive, arcs are not.
Always have a reason for the method you
adopt, although you may not be able to express
it satisfactorily to another. Such, for example,
as this : If from one end of a line, as a centre, I
describe a circle of a certain size, and then from
the other end of the line, as another centre, I
describe another circle of the same size, the
points where those circles intersect each other,
if they intersect at all, must have the same rela
12 TO THE PUPIL
tion to one end of such line which they have to
The most improving method of entering the
solutions is to show, in a first figure, all the cir-
cles in full by which you have arrived at the
solution, and to draw a second figure in ink,
without the circles.
It is not so much the problems which you
are assisted in performing, as the problems you
perform yourself, that will improve your talents
and benefit your character. Refrain, then, from
looking at the constructions invented by other
persons at least till you have discovered a
construction of your own. The less assistance
you seek the less you will require, and the less
you will desire.
As the power to invent is ever varying in
the same person, and as no two persons have
that power equally, it is better not to be anxious
about keeping pace with others. Indeed, all
your efforts should be free from anxiety. Pleas-
urable efforts are the most effective. Be as-
sured that no effort is lost, though at the time
it may appear so You may improve more
TO THE PUPIL. 13
while studying one problem that is rather intri-
cate to you, than while performing several that
are easy. Dwell upon what the immortal New-
ton said of his own habit of study. " I keep,"
says he, " the subject constantly before me, and
wait till the first dawnings open by little and
little into a full and clear light."
THE science of relative quantity, solid, su-
perficial, and linear, is called Geometry, and
the practical application of it, Mensuration.
Thus we have mensuration of solids, mensura-
tion of surfaces, and mensuration of lines ; and
to ascertain these quantities it is requisite that
we should have dimensions.
The top, bottom, and sides of a solid body,
as a cube, 1 are called its faces or surfaces, 1 and
the edges of these surfaces are called lines.
The distance between the top and bottom
of the cube is a dimension called the height,
depth, or thickness of the cube ; the distance
between the left face and the right face is anoth-
1 The most convenient form for illustration is that of the
cubic inch, which is a solid, having equal rectangular surfaces
* A surface is sometimes called a superficies.
16 INVENTIONAL GEOMETRY.
er dimension, called the breadth or width ; and
the distance between the front face and the
back face is the third dimension, called the
length of the cube.
Thus a cube is called a magnitude of three
The three terms most commonly applied to
the dimensions of a cube are length, breadth,
1. Place a cube with one face flat on a table,
and with another face toward you, and say which
dimension you consider to be the thickness,
which the breadth, and which the length.
2. Show to what objects the word height is
more appropriate, and to what objects the word
depth, and to what the word thickness.
As a surface has nc thickness, it has two di-
mensions only, length and breadth. Thus a
surface is called a magnitude of two dimen-
3. Show how many faces a cube has. 1
1 The surfaces of a cube are considered to be plane tup
1NVE8TWNAL GEOMETRY. 17
When a surface is such, that a line placed
anywhere upon it will rest wholly on that sur-;
face, such surface is said to he a plane sur-
As a line has neither breadth nor thickness,
it has one dimension only, that of length.
Thus a line is called a magnitude of one
4. Count how many lines are formed on a
cube by the intersection of its six plane surfaces.
If that which has neither breadth, nor thick
ness, but length only, can be said to have any
form, then a line is such, that if it were turned
upon its extremities, each part of it would keep
its own place in space.
We cannot with a pencil make a line on pa-
per we represent a line.
The boundaries or ends of a line are called
points, and the intersection of two lines gives
As a point has neither length, breadth, no?
1 When the word line is used in these definitions and que*
lions a straight line is always meant
IS INVENTION AL GEOMETRY.
thickness, it is said to have no dimension. It
has position only.
A point is therefore not a magnitude.
5. Name the number of points that are made
by the intersection of the twelve lines of a cube
We cannot with a pencil make a point on
paper we represent a point.
When any two straight lines meet together
from any other two directions than those which
are perfectly opposite, they are said to make an
And the point where they meet is called the
Thus two lines that meet each other on a
cube make an angle.
6. Represent on paper a rectilineal angle.
7. Can two lines meet together without be-
ing in the same plane I
8. Point out two lines on a cube that exist
on the same surface, and yet do not make an
9. Name the number of plane angles on
INVENTWXAL GEOMETRY. 19
the six surfaces of a cube, and the number of
angular points, and say why the angular points
are fewer than the plane angles.
The meeting of two plane surfaces in a line
- for example, the meeting of the wall of a
room with the floor, or the meeting of two of
the surfaces of a cube is called a dihedral
10. Say how many dihedral angles a cube
The corner made by the meeting of three
or more plane surfaces is called a solid angle.
11. Say how many solid angles there are in
When a surface is such that a line, when
resting upon it in any direction, will be touched
by it toward the middle of the line only, and
not at both ends, such surface is called a convex
12. Give an example of a convex surface.
When a surface is such that a line while rest-
ing upon it, in any direction, will be touched by
1 Dihedral means two-surfaced.
20 INVENTION AL GEOMETRY.
it at the ends, and not toward the middle of the
line, such surface is called a concave surface.
13. Give an example of a concave surface.
A simple curve is such, that on being turned
on its extremities, every point along it will
change its place in space; so that, in a simple
curve, no three points are in a straight line.
14. Give an example of a simple curve.
Lines or curves grouped together by way of
illustration, or for ornament, without regard to
magnitude or surface, take the name of dia-
15. Give an example of a diagram.
When a surface l is spoken of with regard to
its form and size, it takes the name of figure.
If the boundaries of a surface are straight
lines, the figure is called a rectilinear figure, and
each boundary is called a side.
Thus we have rectilinear figures of four
aides, of five sides, of six sides, etc.
16. Make a few rectilinear figures.
1 In the definitions and questions of this work, when hi
trord surface is used, a plane surface is meant
GEOMETRY. . %\
When a surface is inclosed by one curve, it
is called a curvilinear figure, and the boundary
is called its circumference.
17. Make a curvilinear figure with one curve
for its boundary, and in it write its name, and
around it the name of its boundary.
18. Make a curvilinear figure with more
than one curve for its boundaries.
A figure bounded by a line and a curve, or
by more lines and more curves than one, ifl
called a mixed figure.
19. Make a mixed figure, having for its
boundaries a line and a curve.
20. Make a mixed figure, having for its
boundaries one line and two curves.
21. Make a mixed figure, having for its
noundaries one curve and two lines.
When a figure has a boundary of such a
form that all lines drawn from a certain point
within it to that boundary are equal to one
another, such figure is called a circle, and such
point is called the centre of that circle ; and
the boundary is called the circumference of the
circle, and the equal lines drawn from the cen-
tre to the circumference are called the radii of
22. Make four circles. On the first write
its name. Around the outside of the second,
write the name of the boundary. In the third,
write against the centre its name. And be-
tween the centre and the circumference of the
fourth circle, draw a few radii and write on each
23. Can you place two circles to touch each
other at 2 particular point ?
24:. Can you place three circles in a row,
and let each circle touch the one next to it ?
A part of the circumference of a circle is
called an arc.
When the circumference of a circle is di
vided into two equal arcs, each arc is called a
All arcs of circles which extend beyond a
aemi-circumference are called greater arcs.
INVENT10KAL GEOMETRY. 23
All arcs of circles that are not so great as a
semi-circumference are called less arcs.
A line that joins the extremities of an arc is
called the chord of that arc.
When two radii connect together any two
points in the circumference of a circle which are
on exactly the opposite sides of the centre, they
make a chord, which is called the diameter of
the circle, and such diameter divides the circle
into two equal segments, 1 which take the name
25. Make a circle, and in it draw two radii
in such a position as to divide it into two equal
parts, and write on each part its specific name.
All segments of a circle which occupy more
lhan a semi-circle are called greater segments.
26. Make a greater segment, and on it write
27. Make a greater segment, and on the out-
side of each of its boundaries write its name,
: The word segment means a piece cut off: thus we have
egments of a line and segments of a sphere, as w e j] ae seg-
ments of a circle.
*4 INVENTIONAL QEOMBT&*.
All segments of a circle that do not contain
so much as a semi-circle are callevl less segmenta
28. Make a less segment, and in it write its
29. Make a less segment, and on the outside
of each of its boundaries write its name.
30. Can you cut from a circle more than one
greater segment ?
31. Can you cut from a circle more than one
less segment J
32. Place two circles so that the circumfer-
ence of each may rest upon the centre of the
other, and show that the curved figure common
to both circles consists of two segments, and
may be called a double segment.
33. In how many ways can you divide a
double segment into two equal and similar
34. In how many ways can you divide a
double segment into four equal and similar
35. Can you make two angles with two lines f
INVENTIONAL GEOMETRY. 25
When two lines are so placed as to make two
angles, one of the lines is said to stand upon
the other, and the angles they thus make are
called adjacent angles.
36. Make two unequal adjacent angles with
When one line stands upon another line, in
such a direction as to make the adjacent angles
equal to one another, then each of these angles
is called a right angle.
37. Make two equal adjacent angles, and in
each angle write its proper name.
Either of the sides of a right angle is said to
be perpendicular to the other ; and the one to
which the other is said to be perpendicular is
called the base.
38. Make a right angle, and against the sides
of the right angle write their respective names.
-? 39. Can you make three angles with two
y 40. Can you make four angles with two lines ?
-x 41. Can you make more than four angles
with two lines ? 3
26 INVJSNTIOJfAL &JSOMTJRT.
42. Can you divide a line into two equal
43. Can you divide an arc into two eqnaJ
You have been told that figures bounded by
lines are called linear figures.
44. Make a linear figure having the fewest
boundaries possible, and in it write its name,
and say why such figure claims that name. 1
When a figure has for its boundaries three
equal lines, it is called an equilateral triangle.*
45. Can you make an equilateral triangle?
46. Can you with three lines make two
angles, three, four, five, six, seven, eight, nine,
ten, eleven, twelve, thirteen ?
47. Can you so place two equilateral trian-
gles that one side of one of them may coincide
with one side of the other ?
48. Can you divide an equilateral triangle
into two parts that shall be equal to each othei
and similar to each other?
1 Triangles are also called trilateral^
" Equilateral triangles are also called tngrau.
INVENT10NAL GEOMETRY. - 37
> 49. Can you draw one line perpendicular to
another line, from a point that is in the line but
not in the middle of it ?
The figure formed by two radii and an arc is
jailed a sector.
When a circle is divided into four equal sec-
tors, each of such sectors takes the name of quad-
-? 50. Divide a circle into four equal sectors,
and write upon each sector its specific name.
51. Make a set of quadrants, and write in
each angle its specific name.
To compare sectors of different magnitudes
with each other, geometricians have found it
3onvenient to imagine every circle to be divided
into three hundred and sixty equal sectors ; and
a sector consisting of the three hundred and six-
tieth part of a circle, they have called a degree.
An arc, therefore, of such a sector is an arc of a
degree ; ' and the angle of such a sector is an
angle of a degree.
1 A degree of a circle is concisely marked thus (1) Thir
ty degree* thus (30). Thirty-five degrees thus (85).
28 INVENTION AL* GEOMETRY.
52. Make a set of quadrants, and write in
each angle bow many degrees it contains.
All angles greater or less than the angle oi
a quadrant are called oblique angles.
When an oblique angle is less than a quad
rantal angle, that is less than a right angle, that
is less than an angle of 90, it is called an acute
53. Make an acute angle.
When an oblique angle has more degrees in
it than 90, and less than 180, it is called an
54. Make an obtuse angle.
55. Make an acute-angled sector.
56. Make an obtuse-angled sector.
When a sector has an arc of 180, the radii
tormdng with each other one straight line, it has
the same claim to be called a sector as it has to
be called a segment, and yet it seldom takes the
name of either, being generally called a semi-
57. Make three sectors, each containing 180,
INVENTION A L GEOMETRY. . 29
nud write in each sector a different name, and
yet an appropriate one.
A sector which has an arc greater than a
semi-circumference is said to have a reentrant
58. Make a reentrant-angled sector.
59. Say to which class of sectors the degree
You have halved a line, and you have halved
60. Can you divide a segment into two parts
that shall be equal to each other, and similar to
each other ?
61. Can you divide a sector into two parts
that shall be equal to each other, and similar to
each other ?
It is said by some, the circumference of a
circle is 3 times its own diameter ; by others,
more accurate, that it is 3^ times its own diain
62. Say how you would determine the ratio
the circumference of a circle bears to its diara
30 *NVENTIONAL GEOMETRY.
eter, and say also what you make the ratio to
You have divided a line, an arc, a segment,
and a sector, into two equal parts.
63. Can you divide an angle into two equal
When a triangle has two only of its sides of
equal length it is called an isosceles triangle,
64. Make an isosceles triangle.
When a triangle has all its sides of different
lengths it takes the name of scalene.
65. Make a scalene triangle.
When a triangle has one of its angles a right
angle, it is called a right-angled triangle.
66. Make a right-angled triangle.
When a triangle has each of its angles less
than a right angle, and all different in size, it is
called a common acute-angled triangle.
67. Make a common acute-angled triangle.
When a triangle has one of its angles obtuse,
it is called an obtuse-angled triangle.
68. Make an obtuse-angled triangle
INVENTIONAL VE'OMETRY. . 31
In describing the properties of a triangle it
is not unusual to mark each angular point of the
triangle with a letter.
Thus the accompanying triangle is called the
triangle A B C, and C
the sides are called
A B, B C, and A C,
and the three angles
are called A, B, C, or the angles C A B, A B C,
69. Can you make an isosceles triangle with-
out using more than one circle ?
When two lines do not meet either way,
though produced ever so far, they are said to be
70. Draw two parallel lines.
71. Can you draw one line parallel to
another, and let the two be an inch apart ?
72. Can you place two equal sectors so that
one corresponding radius of each sector may be
in one- line, and so that their angles may point
the same way ?
1 Of course it means two lines in the same plane.